An overview of the balanced excited random walk

The balanced excited random walk, introduced by Benjamini, Kozma and Schapira in $2011$, is defined as a discrete time stochastic process in $\mathbb Z^d$, depending on two integer parameters $1\le d_1,d_2\le d$, which whenever it is at a site $x\in\mathbb Z^d$ at time $n$, it jumps to $x\pm e_i$ with uniform probability, where $e_1,\ldots,e_d$ are the canonical vectors, for $1\le i\le d_1$, if the site $x$ was visited for the first time at time $n$, while it jumps to $x\pm e_i$ with uniform probability, for $1+d-d_2\le i\le d$, if the site $x$ was already visited before time $n$. Here we give an overview of this model when $d_1+d_2=d$ and introduce and study the cases when $d_1+d_2>d$. In particular, we prove that for all the cases $d\ge 5$ and most cases $d=4$, the balanced excited random walk is transient.

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